Tagged Questions

For questions about groups which have additional structure as a algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure.
Consider using with the (group-theory) tag.

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential ...

From Hilbert 90 (or, more precisely, a generalisation thereof), we know that the first (Galois) cohomology group of $GL_n$ is trivial, no matter the field of definition. However, for unitary groups, ...

I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...

This is a rather embarrassing question, so please let me know of any duplication and I will happily remove it.
I am seeking to understand the $\mathbb Q$-split form of the algebraic group $G_2$, and ...

assume I have a group $G$ over a field of char 0 and $H$ a closed subgroup. When is it true that the group $N(H)/H$ is representable? If $G$ has nice properties, like to be reductive or unipotent is ...

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...

Let me first introduce two definitions:
For a structure $\mathcal{M}$ in a language $\mathscr{L}$ and a subset $X \subseteq M^n$, the fully induced structure on $X$ is a structure $\mathcal{X}$ with ...

I am trying to understand the Borel-Weil theorem, but I am very confused because of the different conventions used in different sources. I am especially confused about two things: (1) the definition ...

Are there any infinite descending chains of closed subgroups of the general linear group over a field? More specifically, is my argument ok? Can you fill in some of the details?
Prop: No.
Proof: If ...

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$.
Let ...

Whenever reading about formal groups, this axiom has always appeared to me as a bit artificial, at least compared to the other axioms.
To explain what I mean, suppose that $R$ is a ring, and that we ...

Given an algebraic group $G$ over an algebraically closed field $K$, if $H$ is a subvariety of $G$, then is $H$ a subgroup of $G$? This seems rather strong. If it is indeed false, is there a geometric ...

Let $f:X\to Y$ be a flat surjective morphism with reduced fibers between affine varieties over an alg. closed field of char. zero, k. Let G be a reductive group acting on X fiberwise.
How do I show ...

In a book I was reading, from the assumptions that we have a linear algebraic group $G$ acting on an irreducible (affine) variety $X$, the author writes that $k(X)^{G}$ is a finitely generated field ...

Recently my professor told me that the usual definition of PSL(2,$\mathbb{R}$) = SL(2,$\mathbb{R}$)/{$\pm$I} does not give an algebraic group, but the following definition does:
PSL(2,$\mathbb{R}$) ...

For an easily comprehension of my questions I write some definitions:
An affine algebraic group scheme over $K$ is a representable group-functor from $K$-algebras category, with a finitely generated ...

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...